Computer-driven American stock markets have become so complex that at any moment in time more than 800 different pricing possibilities are being offered to trading firms across 12 official exchanges, according to new research attempting to explain the tangled system.

The report was prepared by the Royal Bank of Canada, one of the most outspoken critics of the computer-driven American stock market. It will be released to clients this week in advance of a Senate hearing on Thursday that will examine how the structure became so convoluted and what might be done to improve it.

The complexity is a result, in part, of the constant jockeying among exchanges to win business from the biggest traders, many of which are so-called high-frequency trading firms that make money by capitalizing on small changes in prices.

RBC Capital Markets, the division of the bank which led the research, found that the New York Stock Exchange, Nasdaq and other exchanges make frequent small tweaks to their prices, influencing trading behavior in ways that are hard for even sophisticated investors to understand.

Some of the big mutual fund companies that buy and sell stocks on behalf of investors said that before seeing the RBC research even they had not realized how convoluted the system had become.

“The level of complexity has grown to such an extent that it is unknown to most market participants,” said Mehmet Kinak, the head of electronic trading at T. Rowe Price Group, and a client of RBC with which the research has already been shared. “Instead of finding natural buyers and sellers, we’re finding intermediaries who come in and are benefiting from the complexity.”

At one exchange highlighted in the research, EDGX, the pricing tiers, for example, include the “Investor Tier,” the “Ultra Tier,” the “Mega-Step-Up Tier 1” and the “Mega-Step-Up Tier 2,” among others. These tiers, though, are just the first layer of complication that big investors have to navigate to determine the price of what would seem to be a rather simple act: buying or selling a stock.

Many market experts have said that the complexity of the markets is not something that ordinary investors need to worry about. Over the long run, the price of trading stocks has generally gone down as exchanges and trading firms have competed to buy and sell stocks more cheaply.

Press officers for the large exchange companies had no comment on the research, which they had not seen.

But large investment firms like T. Rowe, which handle the retirement savings of millions, have regularly complained that it has become much trickier to confidently trade large blocks of stocks.

Because of the hundreds of pricing structures, the brokers actually buying and selling stocks can be given incentives to make decisions that are not in the best interest of clients like T. Rowe. At a minimum, the opaque pricing structures make it nearly impossible for investors and even brokers to determine what they will pay for a given trade.

“When we trade we don’t even know what it will cost us,” said Rich Steiner, the head of electronic trading strategy at RBC.

Another complaint from critics of the current market structure is that the complexity of the system requires constant software changes that, in turn, increase the chances of technological breakdowns. The trading firm Knight Capital lost hundreds of millions of dollars and sowed panic in the markets in 2012 after installing faulty software in response to rule changes at the exchanges.

The prices are far from the only factor introducing complexity into the markets. Twelve public exchanges are now in operation, compared to a time when the markets were largely ruled by one: the New York Stock Exchange.

Then there are the dozens of so-called dark pools, where stocks can be traded privately away from the public exchanges.

All of these trading venues offer many different types of orders that determine how and when a stock can be traded. A 2014 research report identified 133 unique order types, including some for particular times of the day and others for trades of a particular size.

The new research is likely to strengthen the hand of an upstart company, IEX, that is currently asking regulators for approval to become an official stock exchange.

The founders of IEX, who were the protagonists in the Michael Lewis book “Flash Boys” about high-speed trading, are hoping to provide a more straightforward way to trade stocks. But one tool that IEX is proposing to use — a so-called speed bump to slow down trading — has attracted opposition from existing exchanges and some trading firms.

The chief executive of IEX, Brad Katsuyama, developed many of his ideas when he was working at RBC, and the new research from RBC could be viewed as an effort to help a former employee.

RBC executives have not taken any public position on whether IEX should be given regulatory approval to become an exchange, and IEX is not mentioned in the new RBC report. The report uses the findings to push for a more basic change to the United States stock markets: doing away with the rebates that trading firms can earn for trading at a particular exchange.

RBC and other critics of the stock market structure argue that the rebates given out by exchanges can skew the incentives of brokers and banks, encouraging them to trade where they can get the largest rebate, rather than where they can get the best price for their client.

The pricing structures that RBC details in its new report are a result of the efforts by exchanges to calibrate the rebates they offer to some customers and the fees they charge to others.

In one example given in the report, the BATS-Y stock exchange — one of four stock exchanges run by BATS Global Markets — sent out a fee notice at the end of March 2014 announcing that it would offer 15-hundredths of a cent to traders buying certain stocks, thus bettering the 14-hundredths of a cent that Nasdaq BX had been offering. Fifteen minutes after the BATS-Y filing, Nasdaq made its own filing matching the new BATS-Y price. The next morning, BATS-Y filed again, increasing its offer to 16-hundredths of a cent.

Between 2012 and 2015, RBC found 362 filings with regulators announcing changes to trading fees, with some of the filings including multiple fee changes. The number of pricing tiers proliferates quickly because each tier can apply to similar trades in different ways depending on how frequently a trader uses a particular exchange.

Vimal Patel, who oversaw the research at RBC, said that he had no idea how tangled it had become until he began trying to sketch it out last summer.

“It snuck up on people that the world is this complicated,” he said.

Correction:

An article on March 2 about the complexity of pricing on computer-driven stock markets misstated the price offered by the BATS-Y exchange, one of four stock exchanges run by BATS Global Markets, to traders of certain stocks. The prices are quoted in the hundredths of a cent, not in the thousandths of a cent.

A version of this article appears in print on , on Page B3 of the New York edition with the headline: Stock Exchange Prices Grow So Convoluted Even Traders Are Confused. Order Reprints | Today’s Paper | Subscribe

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